Simplicity of confinement in SU(3) Yang-Mills theory

Abstract

We introduce a novel observable associated to Abelian monopole currents defined in the Maximal Abelian Projection of SU(3) Yang-Mills theory that captures the topology of the current loop. This observable, referred to as the $\textit{simplicity}$, is defined as the ratio of the zeroth over the first Betti number of the current graph for a given field configuration. A numerical study of the expectation value of the simplicity performed in the framework of Lattice Gauge Theories enables us to determine the deconfinement temperature to a higher degree of accuracy than that reached by conventional methods at a comparable computational effort. Our results suggest that Abelian current loops are strongly correlated with the degrees of freedoms of the theory that determine confinement. Our investigation opens new perspectives for the definition of an order parameter for deconfinement in Quantum Chromodynamics able to expose the potentially rich phase structure of the theory.

Figures (17)

• Figure 1: For $N_t = 8$, $\lambda$ and $\chi_\lambda$ as functions of $\beta$ zoomed into the critical region with translucent lines to guide the eye. The vertical line and band show respectively the central value and the statistical error for the extrapolated $\beta_{c}$ determined in Ref. Lucini:2003zr.
• Figure 2: Position of the peak of $\chi_\lambda$ for lattices with $N_t = 4$ and $N_s = 16,20,24,28,32$ (blue triangles). Dashed lines represent the polynomial regression fits used for the infinite volume extrapolation of these peak values, and the red circle is the resulting estimate of $\beta_{c}$ in the thermodynamic limit (see the supplemental material for additional details). The horizontal line and band show respectively the central value and the statistical error for the extrapolated $\beta_{c}$ determined in Ref. Lucini:2003zr.
• Figure 3: Position of the peak of $\chi_\lambda$ for lattices with $N_t = 6$ and $N_s = 24,30,36,42,48$ (blue triangles). Dashed lines represent the polynomial regression fits used for the infinite volume extrapolation of these peak values, and the red circle is the resulting estimate of $\beta_{c}$ in the thermodynamic limit (see the supplemental material for additional details). The horizontal line and band show respectively the central value and the statistical error for the extrapolated $\beta_{c}$ determined in Ref. Lucini:2003zr.
• Figure 4: Position of the peak of $\chi_\lambda$ for lattices with $N_t = 8$ and $N_s = 32,40,48,56,64$ (blue triangles). Dashed lines represent the polynomial regression fits used for the infinite volume extrapolation of these peak values, and the red circle is the resulting estimate of $\beta_{c}$ in the thermodynamic limit (see the supplemental material for additional details). The horizontal line and band show respectively the central value and the statistical error for the extrapolated $\beta_{c}$ determined in Ref. Lucini:2003zr.
• Figure S1: Left: A graph $G$ and a spanning tree $T$ shown in red. Right: the graph $G/T$ resulting from contracting $T$ to a single vertex.